The first one I think is the basic definition and the rest follow from that, some of the proofs are on the wikipedia page, and what do you mean by a vector times a vector? The above equalities allow a general definition of a product of quaternions.

Just like:A=i+j+kB=2i+3j-4k
They are both vector.
I heard that a vector times a vector become a tensor(rank two).
If ij=-k,ik=-j,jk=-i.
The product is:
(i+j+k)(2i+3j-4k)=-2-3k+4j-2k-3+4i-2j-3i+4=i+2j-5k-1
I think it is still a vector, not a tensor(rank two).

The product of two quaternions is still a quaternion (and quaternions are not vectors). You did the multiplication wrong up there, if A and B are as you had then AB is -7i + 6j + k - 1.

You can define many products between vectors, for example in [itex]\mathbb{R}^3[/itex] you have the usual cross and dot products (and the dot product generalizes to other spaces of course). Those two products can be read off the result of quaternion multiplication: